Question

Solve the following question and mark the best possible option.
If a, b, c > 0, then find the minimum value of (a + b + c)`(1/(a + b) + 1/(b + c) + 1/(c + a))`.

Options

• `7/2`

• 4

• `9/2`

• None of these

Answer 1

We have AM ≥ GM
⇒ `(1/(a + b) + 1/(b + c) + 1/(c + a)) ≥ 3 root(3)(1/(a + b) xx 1/(b + c) xx 1/(c + a))`   ...(1)

And (a + b) + (b + c) + (c + a) ≥ 3 `root(3)((a + b)(b + c)(c + a))`
Hence
`{(a + b) + (b + c) + (c + a)} (1/(a + b) + 1/(b + c) + 1/(c + a)) ≥ 3 root(3)(1/(a + b) xx 1/(b + c) xx 1/(c + a)) xx 3 root(3)((a + b)(b + c)(c + a))`

Solving we get `2(a + b + c)(1/(a + b) + 1/(b + c) + 1/(c + a)) ≥ 9`,

⇒ `(a + b + c)(1/(a + b) + 1/(b + c) + 1/(c + a)) ≥ 9/2`

Hence the minimum value is `9/2`.